A "living dex" is a fan designation for when a player completes the Pokédex and has one of each species in the PC or the party at the same time. It's often considered to be much more difficult and time-consuming than a regular Pokédex because it often involves, for instance, playing the same game multiple times to get multiple copies of a Legendary Pokémon.
The method by which you can get a mostly-completed Pokédex on Platinum is described here. This page will focus more on requirements specific to a living dex using only Diamond, Pearl, and Platinum.
You will need access to at minimum three separate Diamond, Pearl, and Platinum save files in order to obtain all starters. Since you can breed them, only three files are required.
This is just to note that if you are attempting to complete a living dex on all of Diamond, Pearl, and Platinum simultaneously, you will need at minimum four save files, in order to ensure that each copy of Diamond, Pearl, and Platinum gets access to Dialga and Palkia. This would require one Diamond, one Pearl, and two Platinum, where the second Platinum file trades Dialga to Pearl and Palkia to Diamond.